Octave 3.8, jcobi/3

Average Error: 0.4 → 0.3

Time: 33.4s

Precision: 64

\[\alpha \gt \left(-1\right) \land \beta \gt \left(-1\right)\]

Math

\[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

↓

\[\frac{\frac{\frac{1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\frac{\beta + \left(\alpha + 2 \cdot 1\right)}{\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\alpha + \beta\right)\right), \beta, \alpha\right)\right), 1.0, 1.0\right)\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

C

double f(double alpha, double beta) {
        double r3275661 = alpha;
        double r3275662 = beta;
        double r3275663 = r3275661 + r3275662;
        double r3275664 = r3275662 * r3275661;
        double r3275665 = r3275663 + r3275664;
        double r3275666 = 1.0;
        double r3275667 = /* ERROR: no posit support in C */;
        double r3275668 = r3275665 + r3275667;
        double r3275669 = 2.0;
        double r3275670 = /* ERROR: no posit support in C */;
        double r3275671 = 1.0;
        double r3275672 = /* ERROR: no posit support in C */;
        double r3275673 = r3275670 * r3275672;
        double r3275674 = r3275663 + r3275673;
        double r3275675 = r3275668 / r3275674;
        double r3275676 = r3275675 / r3275674;
        double r3275677 = r3275674 + r3275667;
        double r3275678 = r3275676 / r3275677;
        return r3275678;
}

↓

double f(double alpha, double beta) {
        double r3275679 = 1.0;
        double r3275680 = alpha;
        double r3275681 = beta;
        double r3275682 = r3275680 + r3275681;
        double r3275683 = 2.0;
        double r3275684 = 1.0;
        double r3275685 = r3275683 * r3275684;
        double r3275686 = r3275682 + r3275685;
        double r3275687 = r3275679 / r3275686;
        double r3275688 = r3275680 + r3275685;
        double r3275689 = r3275681 + r3275688;
        double r3275690 = /*Error: no posit support in C */;
        double r3275691 = /*Error: no posit support in C */;
        double r3275692 = /*Error: no posit support in C */;
        double r3275693 = /*Error: no posit support in C */;
        double r3275694 = r3275689 / r3275693;
        double r3275695 = r3275687 / r3275694;
        double r3275696 = r3275686 + r3275679;
        double r3275697 = r3275695 / r3275696;
        return r3275697;
}

Error

Bits error versus alpha

Bits error versus beta

Derivation

1. Initial program 0.4

   \[\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
2. Using strategy rm

3. Applied introduce-quire 0.4

   \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\left(\frac{\alpha}{\beta}\right)\right)\right)}}{\left(\beta \cdot \alpha\right)}\right)}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

4. Applied insert-quire-fdp-add 0.4

   \[\leadsto \frac{\left(\frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right)\right)}}{\left(1.0\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

5. Applied insert-quire-add 0.4

   \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]
6. Using strategy rm

7. Applied *p16-rgt-identity-expand 0.4

   \[\leadsto \frac{\left(\frac{\left(\frac{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}{\color{blue}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right) \cdot \left(1.0\right)\right)}}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

8. Applied p16-*-un-lft-identity 0.4

   \[\leadsto \frac{\left(\frac{\left(\frac{\color{blue}{\left(\left(1.0\right) \cdot \left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)\right)}}{\left(\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right) \cdot \left(1.0\right)\right)}\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

9. Applied p16-times-frac 0.4

   \[\leadsto \frac{\left(\frac{\color{blue}{\left(\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right) \cdot \left(\frac{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}{\left(1.0\right)}\right)\right)}}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

10. Applied associate-/l* 0.4

    \[\leadsto \frac{\color{blue}{\left(\frac{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(\frac{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}{\left(1.0\right)}\right)}\right)}\right)}}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

11. Simplified 0.3

    \[\leadsto \frac{\left(\frac{\left(\frac{\left(1.0\right)}{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\color{blue}{\left(\frac{\left(\frac{\beta}{\left(\frac{\alpha}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}\right)}{\left(\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\frac{\alpha}{\beta}\right)\right), \beta, \alpha\right)\right), \left(1.0\right), \left(1.0\right)\right)\right)\right)}\right)}}\right)}{\left(\frac{\left(\frac{\left(\frac{\alpha}{\beta}\right)}{\left(\left(2\right) \cdot \left(1\right)\right)}\right)}{\left(1.0\right)}\right)}\]

12. Final simplification 0.3

    \[\leadsto \frac{\frac{\frac{1.0}{\left(\alpha + \beta\right) + 2 \cdot 1}}{\frac{\beta + \left(\alpha + 2 \cdot 1\right)}{\left(\mathsf{qma}\left(\left(\mathsf{qma}\left(\left(\left(\alpha + \beta\right)\right), \beta, \alpha\right)\right), 1.0, 1.0\right)\right)}}}{\left(\left(\alpha + \beta\right) + 2 \cdot 1\right) + 1.0}\]

Reproduce

herbie shell --seed 2019163 
(FPCore (alpha beta)
  :name "Octave 3.8, jcobi/3"
  :pre (and (>.p16 alpha (real->posit16 -1)) (>.p16 beta (real->posit16 -1)))
  (/.p16 (/.p16 (/.p16 (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 beta alpha)) (real->posit16 1.0)) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1)))) (+.p16 (+.p16 (+.p16 alpha beta) (*.p16 (real->posit16 2) (real->posit16 1))) (real->posit16 1.0))))